2. Fundamentals of Electrochemical Impedance Spectroscopy

3. I. Introduction: physics & electrotechnics Definition & properties of impedance Simple RC circuits and their spectra Measurement principles and graphical analyses

4. How to characterize a two-pole (an electrical system)? Case A: a steady state current-voltage curve, I(U), exists at any moment.

5. How to characterize a two-pole (an electrical system)? Case B: the current-voltage curve, I(U) depends also on time, t The system can be characterized through the time-dependence of the current: I(t) – U(t) relations are analyzed. Possibilities: transient response (following a jump or pulse) ac methods, „frequency” response (sinusoidal perturbation) Passive, linear electrical „two-poles”; I(k*U)=k*I(U)) are considered.

6. Operational definition of impedance: Stimulus: U(t)= U ac ∙sin(ωt)Response: I(t) =I ac ∙sin(ωt+ φ) Impedance is defined as Z  (U ac /I ac and φ) (Since I(k*U)=k*I(U)), the U ac /I ac is not dependent on U ac.)

7. Impedance (at one frequency):  is defined as Z  (U ac /I ac and φ), complex number  Z  U ac /I ac · e iφ = Z abs cos (φ) + i· Z abs sin(φ) Euler’s formula with Z abs  U ac /I ac and i  -1  admittance: Y  1/Z(Y abs =1/Z abs and φ Y = -φ Z )  immittance = common term for impedance and admittance

8. Impedance (as function of frequency, (ω=2πf )):  it is called a spectrum (typically 10 -3 /s <  < 10 +7 /s   representations r∙e iφ  r∙(cos(φ)+i∙sin(φ)) = Re + i∙Im; Nyquist: Im(Z) vs Re(Z) ln(r∙e iφ )  ln(r)+i∙φ Bode: lg(Z abs ) vs lg(f ) and φ vs lg(f )

9. Impedance:  of a resistor: Z R  R  of a capacitor: Z C  1/(i  C)  of an inductor: Z L  i  L Impedance of a network of RCL elements can be calculated just in the same way as the resistance of a network of resistors. Impedance:  of serially connected elements: Z s = Z 1 +Z 2 1/Y s = 1/Y 1 +1/Y 2  of parallely connected elements: Y p = Y 1 +Y 2 1/Z p = 1/Z 1 +1/Z 2

10. I. Introduction: physics & electrotechnics Definition & properties of impedance Simple RC circuits and their spectra Measurement principles and graphical analyses

11. How do the spectra look like? Examples of simple circuits:

12. Resistance:

13. Capacitance:

14. Rs-Cs:

15. Semicircle Characteristic frequency ω 0 =1/RpCp Time constant τ 0 =1/ω 0 Cp||Rp

16. Symmetries:

17. Rs-Cp||Rp

18. (Rs-Cs)||Rp

19. Circuits of different topologies may have the same impedance function. There is no unique connection of circuit and spectrum.

20. Rp||Cp sequences yield semicircle sequences; they are merged if the RC time constants are close to each other.

21. I. Introduction: physics & electrotechnics Definition & properties of impedance Simple RC circuits and their spectra Measurement principles and graphical analyses

22. ac voltage source: sine-wave generator Simplest way: using sine-wave voltages FRA, lock-in amplifier Umeas:U(t)= U ac ∙sin(ωt) Imeas:I(t) =I ac ∙sin(ωt+ φ)

23. Typical measurement setups: for resistive systems for capacitive systems (dielectric spectroscopy)

24. Simple way of analysis: plotting and determining characteristic values RsRs Rs+ RpRs+ Rp C p =1/(ω 0 *R p ) Use various representations.

25. Nyquist vs Bode representations: Advantages (both are good): Nyquist: „structures” are better seen Bode: complete documentation of the data ln(r∙e iφ )  ln(r)+i∙φ

26. ac voltage source: sine-wave generator Simplest way: using sine-wave voltages FRA, lock-in amplifier Umeas:U(t)= U ac ∙sin(ωt) Imeas:I(t) =I ac ∙sin(ωt+ φ)

27. Turnkey EIS systems are available for 15-60 k€.

28. Mechanistic studies and identification of processes Goal: Identification of the appropriate model (equivalent circuit AND the underlying physico-chemical processes) Measure Z(ω) as function of E, c i, T, etc Repeat Construct model with reasonable assumptions; calculate its impedance function (perhaps expressed as an equivalent circuit, also as function of E, c i, T, etc) Estimate the model’s parameters (e.g. by NLLS fitting) Until a. the measured and calculated Z(ω) spectra are similar to each other; b. the E, c i, T, etc dependencies are correct (not self-contradictory).

29. Rs: Bulk conductivities General characterization Cp: Interfacial capacitance Structure of the interface (double layer, adsorption) Rp: Charge transfer resistance Electrochemical kinetics Corrosion Bulk - interface Dielectric spectroscopy Determination of values of parameters (when the model is already established)

30. Measurement modes: Multiple frequency mode: impedance spectrum measurement (at constant E dc ) – followed by the determination of the parameters Single frequency mode: with scanned E dc ; examples: ac voltammetry (for characterization of charge transfer); capacitance measurements with (high) f and with (slowly) scanned E dc requires the a priori knowledge of the equivalent circuit;

31. Bulk resistance Resistance is determined through impedance spectrum measurements if: a single resistance cannot be measured (only a network’s impedance); typically high resistance materials which are difficult to be contacted; jointly with the determination of permittivity; polymer membranes, ionic conductors, porous structures.

32. Time evolution of the impedance spectra of a physically drying styrene- acrylate self-standing resin film in 0.1M KNO 3 solution (100 kHz - 1 Hz), Lendvay-Győrik et al, 2007.

33. Interfacial capacitance 2. Determination of „zero points” (of space charge layers): I.Metal / solution of a binary electrolyte of low concentration Hg (Au, Ag) in 1-100 mM NaF solution (two mobile charge carriers) II: Metal / extrinsic semiconductor junction n – or p doped semiconductor metallized or a semiconductor electrode in aqueous solution (one fixed and one mobile charge carrier) thus 1. Calculation of adsorbate coverages:

34. Interfacial capacitance Determination of „zero points” (of space charge layers): Model: The distribution of the mobile charges (ions or electrons or holes) is determined by the electrostatic potential and the thermal motion: Poisson-Boltzmann equation Expressed quantity: space charge layer capacitance vs potential.

35. „Gouy-Chapman minimum”, Hg in NaF solution, Grahame (1947) Determination of „zero points”, A: Metal / solution of a binary electrolyte of low concentration (two mobile charge carriers): Capacitance has a minimum at the pzc (potential of zero charge - at which the ion accumulation nearby the metal, in the solution vanishes).

36. Determination of „zero points”, B: Metal (or electrolyte) / extrinsic semiconductor junction (one fixed and one mobile charge carrier) 1/C 2 vs E: determination of n 0 and E fb (flatband potential – at which the space charge layer in the semiconductor vanishes) Mott-Schottky plot (ZnO, Freund & Morrison, 1989)

37. Parallel resistance Parallel resistance – interpreted as a charge transfer resistance exchange current density is calculated ( = kinetics information) typical use: average corrosion rate is calculated for details, see many application notes

38. Fe in H 2 SO 4 (5..100mM) at corrosion potential (Lendvay-Győrik et al, 2000) TiC x N y film (on steel) in Na 2 SO 4 (0.5M) at function of time (Senna et al, 2000) Determination of R p - corrosion tests

39. Determination of R p - corrosion tests, inhibitor studies Fe in 1M HCl with and without 1 mM oct-1-yn-3-ol (octynol, inhibitor), at corrosion potential (Lendvay-Győrik et al, 2003) a b c a: without 1-octynol b: with 1 mM 1-octynol c: with 1 mM 1-octynol, after an anodic treatment

40. Coating tests: An ideal polymer, insulating coating is capacitive. Corrosion + transport through the pores – causes a shunt term - C||R. EIS response of a pipeline coating in 5% NaCl at 65°C, L. Gray et al (2003) in: D. Loveday et al, JCT CoatingsTech, 2005

41. Technical issues: 2. Decrease noise. Use Faraday-cage. Use the preamplifier supplied with the potentiostat. Connect an oscilloscope to the E output of the potentiostat to monitor noise level. 1. EIS can be used for characterizing stable systems only. A good practice for testing stability is to repeat the measurements (e.g. with decreasing then increasing frequencies). Kramers-Kronig test may help. 3. Troubleshooting: Check the system by measuring the impedance spectra of resistors and dummy cells of similar characteristics to the systems studied.

42. 4. Cells for high frequency (>1..10 kHz) impedance measurements: a.Low impedance reference electrode should be used b.Avoid cells of low „feedback ratio” c.Ensure uniform current density distribution

43. Traditional, „clean” cells may have bad hf behaviour

44. a. Low impedance reference electrode: Avoid high resistance solution paths between cell and reference electrode. To shunt the high resistance paths, use a capacitively coupled auxiliary reference electrode (C≈1-10μF)

45. b. Low feedback ratio: Do not place the counter electrode „far away” from the W and R

46. From www.mpmtechnologies.com, MPM Technologies, Inc., State College, PA, USAwww.mpmtechnologies.com c. Uniform current density distribution

47. 5. Calibration: with dummy cells having Z(ω) similar to the system studied + with two resistors (approx. R sol,1 and R sol,2 ) for the hf accuracy.

48. Fitting of impedance spectrum: a demo; measurement: Ir(100) in 0.1M HCl, 0.1V vs SCE, model: □,◊ measured, x,+ calculated absolute values and phase angles

49. Calculate & plot the difference of the measured and fitted points – try to get rid of the systematic deviations Always plot the measured and fitted curves together in various representations (Bode is the best for this) Inspect errors of parameters

50. 1. Raw data: measured impedance spectra Nyquist (r∙e iφ  r∙(cos(φ)+i∙sin(φ)) = Re + i∙Im): Im(Z) vs Re(Z), „structures” are better seen Scale of Im(Z) and Re(Z) must be identical Bode(ln(r∙e iφ )  ln(r)+i∙φ): lg(|Z|) vs lg(f ) and φ vs lg(f ), for documentation of the data Other representations like log(Im(Z)) vs log(Re(Z)): avoid How to present data? 2. Processed impedance data (with or without fitting) a. Normalize to unit area (Ohmcm 2 ) b. Subtract series (solution) resistance → interfacial Z i c. Z i → interfacial admittance, Y i (ω) → interfacial capacitance, C i (ω) C i (ω)  Y i (ω) /i ω=1/ [i ω(Z(ω)-Z(ω  ))] is also a complex function → Bode, Nyquist d. Plot together fitted and measured spectra

51. VIII. Summary and closing

52. Impedance: perturbation method to study  kinetics (of transport in bulk and of charge transfer across the interface; dk/dE)  interfacial structures (through differential capacitance of the double layer; dq M /dE) What should impedance spectroscopy be used for? Use EIS if the mechanism of the processes are known then numbers (rate coefficients, interfacial charges, diffusion coefficients) can be obtained (determination of corrosion rates); else for testing reaction kinetics hypotheses; endif END.